Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 20, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 20, 2021

11:00 am in Zoom (contact Olivia Beckwith at obeck@illinois.edu for details),Tuesday, April 20, 2021

#### To Be Announced

###### Robert Schneider (University of Georgia)

11:00 am in Zoom,Tuesday, April 20, 2021

#### Exotic $K(h)$-local Picard groups when $2p-1=h^2$ and the Vanishing Conjecture

###### Ningchuan Zhang (University of Pennsylvania)

Abstract: The study of Picard groups in chromatic homotopy theory was initiated by Hopkins-Mahowald-Sadofsky. By analyzing the homotopy fixed point spectral sequence for the $K(h)$-local sphere, they showed that the exotic $K(h)$-local Picard group $\kappa_h$ is zero when $(p-1)\nmid h$ and $2p-1>h^2$. In this joint work in progress with Dominic Culver, we study $\kappa_h$ when $2p-1=h^2$ and show that its vanishing is implied by a special case of Hopkins’ Chromatic Vanishing Conjecture. Goerss-Henn-Mahowald-Rezk defined an algebraic detection map for $\kappa_h$, which is injective in this case. We will use the Gross-Hopkins duality reduce the target of the detection map to some Greek letter element computations. The vanishing of $\kappa_h$ is then implied by some bounds on the divisibility of those Greek letter elements. At height $3$ and prime $5$, the Miller-Ravenel-Wilson computation implies that exotic elements in $\kappa_3$ are not detected by a type 2 complex. The full vanishing of $\kappa_3$ requires a bound on the $v_1$-divisibility of $\gamma$-family elements. Using the same duality argument, we can also reduce the mod-$p$ Homological Vanishing Conjecture to some bounds on the divisibility of Greek letter elements. By comparing the bounds in both cases, we conclude that the Vanishing Conjecture implies $\kappa_h=0$ when $2p-1=h^2$.

For Zoom info, please email vesna@illinois.edu

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, April 20, 2021

#### Geodesic Length in First-Passage Percolation

###### Firas Rassoul-Agha (University of Utah)

Abstract: We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produce singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments. This is joint work with Arjun Krishnan and Timo Seppalainen.

2:00 pm in Zoom,Tuesday, April 20, 2021

#### Isomorphic bisections of cubic graphs

###### Shagnik Das (FU Berlin)

Abstract: Graph partitioning is the division of a graph into two or more parts based on certain combinatorial conditions, and problems of this kind of have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs.

This is joint work with Alexey Pokrovskiy and Benny Sudakov.